Step Of Random Walk Research Articles

Binary theory of impurity quenching accelerated by resonant excitation migration in a disordered system

The excitation quenching on the energy acceptors in the process of non-Markovian random walks throughout a disordered system of donors has been considered. The encounter theory was exploited to obtain the kinetic equations, which are binary to acceptor concentration. Migration of excitation is described by the CTRW model. The quenching kinetics and the rate constants have been calculated in both hopping and diffusion limits. It has been shown that the role of non-Markovian origin of migration becomes the more important, the larger is the length of the random walk step.

Statistics of unentangled polymer loops and primitive paths

AbstractThe concepts of reptation and the tube model have been successfully used to describe the dynamics of a system of entangled polymers. Attempts to apply this model have given rise to questions about the statistics of a polymer, represented by a lattice random walk, and its entanglement with an obstacle net. We have determined the number of ways such a walk can form unentangled closed loops of various types. If one reels in a general random walk from its ends, pulling out unentangled loop, one is left with the so‐called primitive path, which is taken to represent the path of the tube. The probability that an N step random walk has a K step primitive path has been calculated. Asymptotic formulas for this probability are presented.

Universality of the spectral dimension of percolation clusters

We have calculated, via an extensive Monte Carlo simulation, the spectral dimension $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}$ of infinite percolation clusters, at different Euclidean dimensions $2<~d<~6$ and $d=\ensuremath{\infty}$. $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}$ is extracted from the asymptotic behavior of the average number ${S}_{N}$ of distinct visited sites during an $N$-step random walk. Correction to scaling is shown to be very important. For all $d>~2$, ${S}_{N}$ takes the following form: ${S}_{N}=a{N}^{s}(\frac{1+b{N}^{\ensuremath{-}\ensuremath{\omega}}}{a})$, where $s=\frac{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}{2}$, $0.661<~s<~0.666$, and $\ensuremath{\omega}\ensuremath{\simeq}\frac{1}{6}$. We compare our results with existing theories or conjectures relative to the value of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}$.

Statistics of the entanglement of polymers: Unentangled loops and primitive paths

Attempts to apply the reptation and tube model to describe polymer disentanglement and associated relaxations have had wide success. However, questions have arisen in various applications as to the statistics of entanglement of a random walk with an obstacle net. Calculations are presented of a number of probabilities which characterize the degree to which a polymer, represented by a lattice walk, entangles with an array of obstacles. In particular, we have calculated the number of ways that such a walk can form unentangled closed loops of various types. If one reels in a general random walk from its ends, pulling out unentangled loops, one is left with the so-called primitive path, which is taken to represent the path of the tube. The probability that an N step random walk has a K step primitive path has been determined. Asymptotic formulas for this probability are presented.

Anomalous Diffusion on Percolating Clusters

The mean square distance reached after $t$ random-walk steps on a percolating cluster is shown to behave as ${t}^{\frac{2}{(2+\ensuremath{\theta})}}$ (instead of $t$) for short times (for which the cluster is seen to be self-similar). The exponent $\ensuremath{\theta}$ is related to that describing the dc conductivity near percolation. Averaging over all clusters yields the ac conductivity and dielectric constant near percolation in the high-frequency limit, when the polarization of the medium is unimportant.

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.

Isobaric versus Canonical Ensemble Formalism for Monte Carlo Studies of Liquids

The Metropolis Monte Carlo method is one of two main approaches to computer simulation of liquid properties. It has virtually always been employed within the canonical ensemble formalism. By including density as one of the variables for the random walk, the Metropolis method becomes applicable to the analysis of a closed, isothermal, isobaric system (isobaric ensemble conditions). The analysis is directed toward equilibrium properties of classical models of dense polyatomic liquids such as water. Density, compressibility, constant pressure heat capacity, enthalpy and coefficient of thermal expansion are obtained directly in terms of mean values and variances of a two-dimensional distribution, g(U, V) of random walk steps. The method appears to be well suited to the study of liquids in the vicinity of the triple point.

Span of a Polymer Chain

The x span of a polymer chain is defined as the difference between the largest x coordinate and the smallest x coordinate of any segments in the chain. The polymer chain is represented by a nearest-neighbor lattice-model random walk in which the mean square displacement of each component of a single step is 13. The distribution function of the x span of an N step random walk is obtained; and the following asymptotic formulas are obtained for its first and second moments, respectively, 2(2N/3π)1/2 and 4 ln2 N/3. The corresponding moments of the magnitude of the x component of the end-to-end distance are (2N/3π)1/2 and N/3. Excluded volume effects are not considered. It is noted that the problem of calculating the first moment of the x span is identical with the one-dimensional case of the Dvoretzky-Erdös problem, namely, the calculation of the average number of different lattice sites visited in an N-step random walk on a d-dimensional lattice.

Statistical Roughness Parameter as Indicator of Channel Flow Resistance

In an analysis of field data of four natural rivers a statistical, large‐scale roughness parameter has proved effective as an indicator of the channel resistance to the flow when the Froude number based on the mean annual flow is larger than about 0.2. The parameter is derived from the concept that the irregular shape and slope of an irregular alluvial channel may be considered as a result of the random walk of imaginary ‘particles’ along the channel bed. It is the so‐called diffusion coefficient which, according to the stochastic theory of the diffusion process, is normally defined in a diffusion equation as the mean square or the variance of the displacement of the particles during a step of random walk over a constant time interval of length Δt divided by 2 Δt. In the present study, however, Δt represents a constant distance interval instead. The parameter proved to have characteristics similar to those of another large‐scale roughness parameter that is derived by other investigators from concepts other than the statistical. This roughness parameter provides a new statistical geometric measure of the irregularities along the entire channel bed and sides that is related to the probability of channel geometry through the diffusion equation, which provides the conceptual basis for its use as a roughness parameter.

Sequential Analysis, Direct Method

This paper describes from first principles the direct calculation of the operating characteristic function, O.C., the probability of accepting the hypothesis θ = θ0, and the average sample size, A.S.N., required to terminate the test, for any truncated sequential test once the acceptance, rejection, and the continuation regions are specified at each stage. What is needed is to regard a sequential test as a step by step random walk, which is a Markov chain. The method is contrasted with Wald's and two examples are included.

Nuclear Spin Relaxation by Translational Diffusion in Solids

Expressions for ${T}_{1}$ and ${T}_{2}$ have been derived for dipolar relaxation via atomic diffusion using the general theory of nuclear spin relaxation. General methods of evaluating the autocorrelation functions of various terms in the dipole-dipole Hamiltonian are discussed, and cubic symmetry requirements are given. A random walk model is used for the calculation, but rough estimates are made of the effects of correlation in direction and time of successive jumps of atoms for the vacancy mechanism of diffusion; if account were taken of these correlations, the derived relaxation times might change by a factor of nearly 2. Detailed computations are made only in the limits of high and low field. The random walk model yields an expression for ${T}_{1}$ in the high-field limit identical to that given by Torrey. Zero-frequency spectral densities needed for computation of ${T}_{2}$, and also ${T}_{1}$ at low field, are expressed as lattice sums involving only the dipolar interaction and the sum of the probabilities of $n$-step random walks between lattice points (an extension of the Polya problem). Detailed computations of ${T}_{1}$ and ${T}_{2}$ have been made for two or more species of spins diffusing on an NaCl or fcc lattice. The angular dependence of ${T}_{1}$ and ${T}_{2}$ may be large for the NaCl lattice in the high-field limit. The agreement with Torrey's theory for ${T}_{1}$ in the fcc lattice is good.